{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:JWGUDMY2EZTGCIWFQBHUNHDNDB","short_pith_number":"pith:JWGUDMY2","canonical_record":{"source":{"id":"1103.0370","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-02T09:26:20Z","cross_cats_sorted":[],"title_canon_sha256":"52f8b375472abd7cbee3bd41d1451b260f3dbdaece27c9f74aa0804b3dee2c01","abstract_canon_sha256":"2fca2e925c0a3c3b16b6426a3894f4f3e79fd58514d1a31c4f577c1624c4cb30"},"schema_version":"1.0"},"canonical_sha256":"4d8d41b31a26666122c5804f469c6d184667d86d80eb09b035fb09605692efc5","source":{"kind":"arxiv","id":"1103.0370","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.0370","created_at":"2026-05-18T04:22:10Z"},{"alias_kind":"arxiv_version","alias_value":"1103.0370v4","created_at":"2026-05-18T04:22:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0370","created_at":"2026-05-18T04:22:10Z"},{"alias_kind":"pith_short_12","alias_value":"JWGUDMY2EZTG","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JWGUDMY2EZTGCIWF","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JWGUDMY2","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:JWGUDMY2EZTGCIWFQBHUNHDNDB","target":"record","payload":{"canonical_record":{"source":{"id":"1103.0370","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-02T09:26:20Z","cross_cats_sorted":[],"title_canon_sha256":"52f8b375472abd7cbee3bd41d1451b260f3dbdaece27c9f74aa0804b3dee2c01","abstract_canon_sha256":"2fca2e925c0a3c3b16b6426a3894f4f3e79fd58514d1a31c4f577c1624c4cb30"},"schema_version":"1.0"},"canonical_sha256":"4d8d41b31a26666122c5804f469c6d184667d86d80eb09b035fb09605692efc5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:10.957394Z","signature_b64":"++q6CalbtIo/3LZ/Pgp7EJNq9SeW7I7/Y9afIyTi4i1b40bXWpzJ4mx9jyamM5sQv91fneOR/Rm7KRPx1fDEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d8d41b31a26666122c5804f469c6d184667d86d80eb09b035fb09605692efc5","last_reissued_at":"2026-05-18T04:22:10.956905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:10.956905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.0370","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:22:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G5vC5TDIc/xoo3SdU8gTLNjgeq3ZYcgzha+i1gGRkOb11vHD/1tgcHT1zmzN1wya4T//UX+Zj+nSEytILNbsAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T02:30:00.787643Z"},"content_sha256":"07a59d3264fc350701a9370614fc8169a4146c44705e3ea95297accea5d76c54","schema_version":"1.0","event_id":"sha256:07a59d3264fc350701a9370614fc8169a4146c44705e3ea95297accea5d76c54"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:JWGUDMY2EZTGCIWFQBHUNHDNDB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"When are two Dedekind sums equal?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sinai Robins, Stanislav Jabuka, Xinli Wang","submitted_at":"2011-03-02T09:26:20Z","abstract_excerpt":"A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \\ | \\ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind-Rademacher sums as well, namely that for any fixed non-negative integer $n$, a positive integer modulus $b$, and two integers $a_1$ and $a_2$ that are relatively prime to $b$, the hypothesis $r_n (a_1,b)= r_n (a_2,b)$ implies that\n  $b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0370","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:22:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dT531HmFDzNzObGQErHY8CDLZiGVJya90IjrCfiwAL6hPT/AgIgU/dpKGjX3/h8YAPyWgDU3bhVIOTJL+HsLBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T02:30:00.788141Z"},"content_sha256":"cd38afb728fb5ce7bb95d677579e13ad20bc9a2a0d613a3b6a1724534018ca87","schema_version":"1.0","event_id":"sha256:cd38afb728fb5ce7bb95d677579e13ad20bc9a2a0d613a3b6a1724534018ca87"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB/bundle.json","state_url":"https://pith.science/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T02:30:00Z","links":{"resolver":"https://pith.science/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB","bundle":"https://pith.science/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB/bundle.json","state":"https://pith.science/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JWGUDMY2EZTGCIWFQBHUNHDNDB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JWGUDMY2EZTGCIWFQBHUNHDNDB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2fca2e925c0a3c3b16b6426a3894f4f3e79fd58514d1a31c4f577c1624c4cb30","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-02T09:26:20Z","title_canon_sha256":"52f8b375472abd7cbee3bd41d1451b260f3dbdaece27c9f74aa0804b3dee2c01"},"schema_version":"1.0","source":{"id":"1103.0370","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.0370","created_at":"2026-05-18T04:22:10Z"},{"alias_kind":"arxiv_version","alias_value":"1103.0370v4","created_at":"2026-05-18T04:22:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0370","created_at":"2026-05-18T04:22:10Z"},{"alias_kind":"pith_short_12","alias_value":"JWGUDMY2EZTG","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JWGUDMY2EZTGCIWF","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JWGUDMY2","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:cd38afb728fb5ce7bb95d677579e13ad20bc9a2a0d613a3b6a1724534018ca87","target":"graph","created_at":"2026-05-18T04:22:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \\ | \\ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind-Rademacher sums as well, namely that for any fixed non-negative integer $n$, a positive integer modulus $b$, and two integers $a_1$ and $a_2$ that are relatively prime to $b$, the hypothesis $r_n (a_1,b)= r_n (a_2,b)$ implies that\n  $b","authors_text":"Sinai Robins, Stanislav Jabuka, Xinli Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-02T09:26:20Z","title":"When are two Dedekind sums equal?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0370","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:07a59d3264fc350701a9370614fc8169a4146c44705e3ea95297accea5d76c54","target":"record","created_at":"2026-05-18T04:22:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2fca2e925c0a3c3b16b6426a3894f4f3e79fd58514d1a31c4f577c1624c4cb30","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-02T09:26:20Z","title_canon_sha256":"52f8b375472abd7cbee3bd41d1451b260f3dbdaece27c9f74aa0804b3dee2c01"},"schema_version":"1.0","source":{"id":"1103.0370","kind":"arxiv","version":4}},"canonical_sha256":"4d8d41b31a26666122c5804f469c6d184667d86d80eb09b035fb09605692efc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d8d41b31a26666122c5804f469c6d184667d86d80eb09b035fb09605692efc5","first_computed_at":"2026-05-18T04:22:10.956905Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:22:10.956905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"++q6CalbtIo/3LZ/Pgp7EJNq9SeW7I7/Y9afIyTi4i1b40bXWpzJ4mx9jyamM5sQv91fneOR/Rm7KRPx1fDEBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:22:10.957394Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.0370","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07a59d3264fc350701a9370614fc8169a4146c44705e3ea95297accea5d76c54","sha256:cd38afb728fb5ce7bb95d677579e13ad20bc9a2a0d613a3b6a1724534018ca87"],"state_sha256":"3b553f9a66554dfc749e6864522aeefd9c15bc0cf2af41c81aaab05034d76cd6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yv84lB41eBg4IgoUzWpi9cY9Dgm3+ucf4iSf3xsb3hBRxJsEirQdtwD9RDxYv7hnxafgMFZrAxAwp5zixzlWCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T02:30:00.791427Z","bundle_sha256":"eecb2b1e786b9feabf5ed0b3f962fc0afa4e44ceb66b147972caa58acdc7e9d8"}}